Symbolism, Imagery, Allegory
For most of the novel, the whole squirrel thing is kind of like a fun game. Sort of like Where's Waldo. But then there's the dinner party. The conversation that brings it all together is kind of intense, so we'll just quote it here for you.
Margaret Thayer admired it in her turn, and said that when she was a child, she imagined Cinderella's glass shoes to be exactly of that greenish blue tint; whereupon Professor Pnin remarked that, primo, he would like everybody to say if contents were as good as container, and, secundo, that Cendrillon's shoes were not made of glass but of Russian squirrel fur—vair, in French. It was, he said, an obvious case of the survival of the fittest among words, verve being more evocative than vair which, he submitted, came not from varius, variegated, but from veveritsa, Slavic for a certain beautiful, pale, winter-squirrel fur, having a bluish, or better say sizily, columbine, shade—from columba, Latin for "pigeon," as somebody here well knows—so you see, Mrs. Fire, you were, in general, correct.
'The contents are fine,' said Laurence Clements.
'This beverage is certainly delicious,' said Margaret Thayer.('I always thought 'columbine' was some sort of flower,' said Thomas to Betty, who lightly acquiesced.) (6.8.9)
Got it? What Pnin is saying is that Cinderella's shoe was probably actually made out of bluish squirrel fur, not glass. This would just be another moment of Pnin spouting random information if it wasn't for the bowl that Victor sent him. Victor's bowl is supposed to be the exact same blue as Cinderella's shoes. It's made out of glass, not fur, but thanks to Pnin we see that it still has some kind of connection to our squirrel friends.
If squirrels mean something terrible is happening to Pnin, this gorgeous blue bowl probably means that something really bad is about to happen. And it does. After the bowl appears, Pnin learns that he has lost his job and the home he thinks he's about to move into. After this whole fiasco, Pnin is washing the bowl and appears to have broken it with a nutcracker. However, miraculously, the bowl is intact. The other facts of his life, however, not so much.
So what does that mean? We would assume the bowl would break, being a sign of terrible things after all. But it doesn't. So does that mean good things are in the future for Pnin? Or does it mean the worst is yet to come? Or that it's his only comfort in a world of woe? Or something totally different?
What we do know is that Nabokov originally wanted to kill Pnin at the end of the novel, but he wasn't allowed to (by his editors). So maybe the bowl surviving its destruction symbolically represented Pnin surviving his intended literary death.